Wavefunction of Spin One-Half Particle

The state of a spin one-half particle is represented as a vector in ket space.
Let us suppose that this space is spanned by the basis kets
. Here,
denotes a
simultaneous eigenstate of the position operators
,
,
, and
the spin operator
, corresponding to the eigenvalues
,
,
,
and
, respectively. The basis kets are assumed to
satisfy the completeness relation

(431)

It is helpful to think of the ket
as the product
of two kets--a position space ket
, and
a spin space ket
. We assume that such a product obeys
the commutative and distributive axioms of multiplication:

(432)

(433)

(434)

where the
's are numbers. We can give meaning to any
position space operator (such as
) acting on the product
by assuming that it operates only on the
factor, and commutes with the
factor.
Similarly, we can give a meaning to any spin operator (such as
) acting
on
by assuming that it operates only on
, and
commutes with
. This implies that every position
space operator
commutes with every spin operator. In this manner, we can give
meaning to the equation

(435)

The multiplication in the above equation is of a quite different type to
any that we have encountered previously. The ket vectors
and
lie in two completely separate vector spaces, and their product
lies in a third vector space.
In mathematics, the latter space
is termed the product space of the former spaces, which are
termed factor spaces. The number of
dimensions of a product space is equal to the product of the number of dimensions
of each of the factor spaces. A general ket of the product space is not
of the form (435), but is instead a sum or integral of kets of this form.

A general state
of a spin one-half particle is represented as a ket
in the product of the spin and position spaces.
This state can be completely specified by two wavefunctions:

(436)

(437)

The probability of observing the particle in the region
to
,
to
, and
to
, with
is
. Likewise,
the probability of observing the particle in the region
to
,
to
, and
to
, with
is
.
The normalization condition for the wavefunctions is